It is difficult to model a critical parameter needed for accurately and efficiently guiding an apparatus along a desired path. One problem is the variety and complexity of the parameters that are needed for effective guidance. Depending on the application, it may be necessary to model elevation, wind velocity, magnetic flux, texture, distortion, or many others. For some guidance systems, it may be necessary to model two or more parameters simultaneously. Some systems have two or more sensors able to measure a parameter real time, while others may have to make several passes with a sensor over a given area to make effective measurements. Some pertinent parameters vary rapidly with time, and some are prone to false measurements.
In practice, it is difficult to avoid the implicit assumption that a parameter of interest will be smoothly-varying, in some sense. For example, suppose that a parameter is measured at a given set of locations, the measurements and their respective locations forming a set of “basis points.” For guessing a parameter's value at locations at which measurements were not made, it is common to model the parameter's value as the nearest one of the basis points. This form of modeling is only valid for parameters that are believed to be smoothly-varying.
Consider systems in which one or more sensors move in relation to a frame of reference that the sensor(s) can detect. In most cases, one or more of the necessary resources are limited. Data storage space, computational power, the number of accessible sensors, measurement time, and precision are all limited resources. For guidance systems of this type, it is a shortcoming of prior parametric modeling systems that very few use lateral profiles that are not piecewise-linear. This shortcoming of prior systems causes effective modeling in this context to be unduly wasteful and inaccurate.